Logic is the philosophical study of valid reasoning. Mathematical logic is an extension of symbolic logic (which is extension of formal logic) into other areas, in particular to the study of model theory, proof theory, set theory, and recursion theory.

How to explain, that logic is the only correct way of describing any valid process? Why it is not possible to define logic in different way, and then build an alternative foundation for systems, processes?

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There are other ways of describing valid processes, not just logical, and they are not "incorrect". You need to specify your goal in such an endeavour before making such statements. If your goal is to say that valid reasoning is the only way to reason with validity about systems and processes, well, I suppose that logically follows. Much as I enjoy mathematics and mathematical logic, I prefer to live in a richer world where one can do more with systems than just reason about them in a formalizable fashion. Gerhard "Ask Me About System Design" Paseman, 2012.03.20
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Gerhard PasemanMar 21 '12 at 6:07

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Classical logic includes within it computability, and computability includes all the processes that we currently know how to construct.

By "includes within it", I mean that if you have some process for coming to a conclusion about some question, and that process fits into one of the standard models of computation (and therefore all of them), you can describe it within logic and then logically reason your way to an outcome. You can prove: "This process starts with the data [abc], and follows rules [xyz], so at the next step it have the data [def], so ......." and end with "this process terminates with the conclusion [uvw]." Thus, conclusions made by any other system can be embedded into conclusions made by logic.

Since we do not know how to build any non-computable system, logic is sufficient. If you come up with some kind of strange non-logical system, a logician can use logic to describe it and determine its conclusions perfectly.